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For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of p-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We prove a number of such supercongruences by using classical hypergeometric transformation formulae. These formulae (see the appendix), most of which are decades or centuries old, allow us to write the terminating series as the ratio of products of of Γ-values. At this point sums have become quotients. Writing these Γ-quotients as Γp-quotients, we are in a situation that is well-suited for proving p-adic congruences. These Γp-functions can be p-adically approximated by their Taylor series expansions. Sometimes there is cancelation of the lower order terms, leading to stronger congruences. Using this technique we prove, among other things, a conjecture of Kibelbek and a strengthened version of a conjecture of van Hamme.

Consider a characteristic p representation of the absolute Galois group of the rational numbers. In this paper we show how to deform this representation to the p -adics while guaranteeing that the characteristic polynomials of Frobenius at a density one set of primes are algebraic and pure of specified weight. The resulting representation is ramified at an infinite (density zero) set of primes. As a consequence of the technique of proof we show that one can compatibly lift a mod pq representation. Again, the resulting representation is ramified at infinitely many primes.

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